Sharp bounds on enstrophy growth in the viscous Burgers equation
Abstract
We use the Cole--Hopf transformation and the Laplace method for the heat equation to justify the numerical results on enstrophy growth in the viscous Burgers equation on the unit circle. We show that the maximum enstrophy achieved in the time evolution is scaled as $\mathcal{E}^{3/2}$, where $\mathcal{E}$ is the large initial enstrophy, whereas the time needed for reaching the maximal enstrophy is scaled as $\mathcal{E}^{-1/2}$. These bounds are sharp for sufficiently smooth initial conditions.
- Publication:
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Proceedings of the Royal Society of London Series A
- Pub Date:
- November 2012
- DOI:
- 10.1098/rspa.2012.0200
- arXiv:
- arXiv:1204.3905
- Bibcode:
- 2012RSPSA.468.3636P
- Keywords:
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- Mathematics - Analysis of PDEs
- E-Print:
- 12 pages